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Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.

Franck Barthe, Patrick Cattiaux, Cyril Roberto (2006)

Revista Matemática Iberoamericana

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the...

Inverse Limit Spaces Satisfying a Poincaré Inequality

Jeff Cheeger, Bruce Kleiner (2015)

Analysis and Geometry in Metric Spaces

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces,...

Józef Marcinkiewicz (1910-1940) - on the centenary of his birth

Lech Maligranda (2011)

Banach Center Publications

Józef Marcinkiewicz’s (1910-1940) name is not known by many people, except maybe a small group of mathematicians, although his influence on the analysis and probability theory of the twentieth century was enormous. This survey of his life and work is in honour of the 100 t h anniversary of his birth and 70 t h anniversary of his death. The discussion is divided into two periods of Marcinkiewicz’s life. First, 1910-1933, that is, from his birth to his graduation from the University of Stefan Batory in Vilnius,...

Korn's First Inequality with variable coefficients and its generalization

Waldemar Pompe (2003)

Commentationes Mathematicae Universitatis Carolinae

If Ω n is a bounded domain with Lipschitz boundary Ω and Γ is an open subset of Ω , we prove that the following inequality Ω | A ( x ) u ( x ) | p d x 1 / p + Γ | u ( x ) | p d n - 1 ( x ) 1 / p c u W 1 , p ( Ω ) holds for all u W 1 , p ( Ω ; m ) and 1 < p < , where ( A ( x ) u ( x ) ) k = i = 1 m j = 1 n a k i j ( x ) u i x j ( x ) ( k = 1 , 2 , ... , r ; r m ) defines an elliptic differential operator of first order with continuous coefficients on Ω ¯ . As a special case we obtain Ω u ( x ) F ( x ) + ( u ( x ) F ( x ) ) T p d x c Ω | u ( x ) | p d x , ( * ) for all u W 1 , p ( Ω ; n ) vanishing on Γ , where F : Ω ¯ M n × n ( ) is a continuous mapping with det F ( x ) μ > 0 . Next we show that ( * ) is not valid if n 3 , F L ( Ω ) and det F ( x ) = 1 , but does hold if p = 2 , Γ = Ω and F ( x ) is symmetric and positive definite in Ω .

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